Mathematical Methods — Question 19

QCAA 2021, Paper 1 · 4 marks

Q19 · 2021 · Technology-freeComplex unfamiliar4 marks

QUESTION 19 (4 marks)

A firm aims to have 95% confidence in estimating the proportion of office workers who respond to an email in less than an hour to within

\pm0.05

. A survey has never been undertaken before, so no past data is available. The firm believes that if the proportion is 0.5, then this will result in the largest variability in the sample proportion. Based on this, determine the sample size needed using the approximate value of

z=2

for the 95% confidence interval. Justify the choice of 0.5 for the proportion.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

n=400

;

\hat p=0.5

maximises

\hat p(1-\hat p)

Worked solution

Using

\text{interval margin}=z\sqrt{\frac{\hat p(1-\hat p)}{n}},

substitute

0.05=2\sqrt{0.5(1-0.5)/n}

to obtain

n=400

. The largest sample size results when

\hat p(1-\hat p)

is maximised; this occurs at

\hat p=0.5

correctly selects the interval margin formula

[1 mark]

substitutes values into the formula

[1 mark]

determines sample size n

[1 mark]

verifies the firm’s decision to use p-hat = 0.5 using mathematical reasoning

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (4 marks)

QCAA guide · typeset solution

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